If \(u\)  is smooth with finitely many zeroes
in \(U\) , the index as defined in (REF )
agrees with (REF ). Indeed, take an orthonormal,
positively oriented tangent frame \(\lbrace \tau _1, \, \tau _2\rbrace \) 
defined in a neighbourhood of \(\partial U\) .
(Such a frame exists, see e.g. [1]}.)
By lifting results, we may write \(u\)  as
\(u = \rho \left((\cos \theta )\tau _1 + (\sin \theta )\tau _2\right)\qquad \textrm {on } \partial U,\) 
